Group #146239 — subgroups

Subgroups — #146239

Order 512.

289 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 289 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13209	C₂	1	2	yes	maximal
2	#12910	C₂	1	2	yes	maximal, complemented
3	#9862	C₂	1	1	yes	maximal, characteristic
4	#8826	C₂	1	1	yes	maximal, characteristic, complemented
5	#7137	C₂	1	1	yes	maximal, characteristic, complemented
6	#3147	C₂²	1	1	yes	characteristic
7	C₂×#368	—	2	4	no	—
8	C₂×#359	—	2	4	no	—
9	C₂×#346	C₂²	1	2	yes	—
10	#1848	C₂²	1	1	yes	characteristic
11	#1848	C₂²	1	2	yes	—
12	#1567	—	2	2	no	—
13	#1492	—	2	2	no	—
14	C₂×#322	C₂²	1	1	yes	characteristic
15	C₂×#135	—	2	2	no	—
16	#532	—	2	1	no	—
17	C₂²×C₁₆	D₄	1	4	yes	—
18	#495	—	2	2	no	—
19	#494	Q₈	1	1	yes	characteristic
20	#493	Q₈	1	1	yes	characteristic
21	#492	—	2	1	no	—
22	C₂×#108	—	2	2	no	—
23	C₂×#108	—	2	2	no	—
24	C₂×#108	—	2	2	no	—
25	#423	—	2	2	no	—
26	#423	—	2	1	no	—
27	C₂×C₄×C₈	D₄	1	2	yes	—
28	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
29	#368	—	4	8	no	—
30	#359	—	4	8	no	—
31	#346	—	2	2	no	—
32	#346	—	2	2	no	—
33	#335	—	4	4	no	—
34	D₄⋊C₈	—	4	4	no	—
35	#322	—	2	2	no	—
36	C₂×#41	—	4	1	no	—
37	#135	—	4	4	no	—
38	#135	—	4	4	no	—
39	C₂×Q₁₆	—	2	2	no	—
40	C₂×SD₁₆	—	4	2	no	—
41	C₂×D₈	—	2	2	no	—
42	C₂×M₁₆	—	4	2	no	—
43	C₂×M₁₆	—	4	1	no	—
44	C₂²×C₈	—	2	2	no	—
45	C₂²×C₈	C₂×D₄	1	2	yes	—
46	C₄⋊Q₈	—	2	1	no	—
47	#127	—	2	1	no	—
48	#124	—	4	1	no	—
49	#121	—	4	2	no	—
50	C₄×D₄	—	4	2	no	—
51	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
52	C₂×C₁₆	—	2	4	no	—
53	C₂×C₁₆	—	2	4	no	—
54	C₂×C₁₆	D₈	1	8	yes	—
55	#108	—	4	4	no	—
56	#108	—	4	4	no	—
57	#108	—	4	4	no	—
58	C₄⋊C₈	—	4	2	no	—
59	C₄⋊C₈	—	4	4	no	—
60	C₄×C₈	—	2	2	no	—
61	C₄×C₈	D₈	1	4	yes	—
62	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
63	C₄×C₈	#41	1	1	yes	characteristic
64	C₄×C₈	#41	1	2	yes	—
65	#41	—	8	2	no	—
66	#41	—	8	2	no	—
67	C₂×Q₈	—	4	1	no	—
68	C₂×D₄	—	4	1	no	—
69	C₂×D₄	—	4	2	no	—
70	C₂×D₄	—	8	1	no	—
71	C₂²×C₄	—	8	1	no	—
72	C₂²×C₄	#127	1	1	yes	characteristic
73	C₂²×C₄	#137	1	2	yes	—
74	Q₁₆	—	4	4	no	—
75	SD₁₆	—	8	4	no	—
76	D₈	—	4	4	no	—
77	M₁₆	—	8	2	no	—
78	M₁₆	—	8	2	no	—
79	M₁₆	—	8	2	no	—
80	C₂×C₈	—	2	4	no	—
81	C₂×C₈	—	8	2	no	—
82	C₂×C₈	—	8	1	no	—
83	C₂×C₈	—	8	2	no	—
84	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
85	C₂×C₈	#123	1	2	yes	—
86	C₂×C₈	C₂×D₈	1	4	yes	—
87	C₂×C₈	#137	1	4	yes	—
88	C₄⋊C₄	—	4	2	no	—
89	C₂²⋊C₄	—	8	2	no	—
90	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
91	C₄²	#123	1	1	yes	characteristic
92	C₄²	C₂×D₈	1	2	yes	—
93	C₁₆	—	2	8	no	—
94	C₂³	—	8	1	no	—
95	C₂³	#497	1	1	yes	characteristic
96	Q₈	—	8	2	no	—
97	D₄	—	8	4	no	—
98	D₄	—	8	2	no	—
99	D₄	—	16	2	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	8	1	no	—
102	C₂×C₄	—	16	1	no	—
103	C₂×C₄	—	16	1	no	—
104	C₂×C₄	#399	1	1	yes	characteristic
105	C₂×C₄	#474	1	2	yes	—
106	C₂×C₄	#480	1	2	yes	—
107	C₂×C₄	#493	1	2	yes	—
108	C₂×C₄	#493	1	1	yes	characteristic
109	C₂×C₄	#497	1	2	yes	—
110	C₈	—	4	4	no	—
111	C₈	—	16	1	no	—
112	C₈	—	16	2	no	—
113	C₈	#474	1	4	yes	—
114	C₈	#480	1	4	yes	—
115	C₂²	—	2	2	no	—
116	C₂²	—	16	1	no	—
117	C₂²	—	16	2	no	—
118	C₂²	#1722	1	2	yes	—
119	C₂²	#2093	1	1	yes	characteristic
120	C₄	—	2	2	no	—
121	C₄	—	2	2	no	—
122	C₄	—	16	1	no	—
123	C₄	#1712	1	1	yes	characteristic
124	C₄	#1722	1	1	yes	characteristic
125	C₄	#2093	1	2	yes	—
126	C₂	—	2	2	no	—
127	C₂	—	32	1	no	—
128	C₂	#10673	1	1	yes	characteristic
129	C₂	#13532	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13209

C₂

yes

maximal

#12910

C₂

yes

maximal, complemented

#9862

C₂

yes

maximal, characteristic

#8826

C₂

yes

maximal, characteristic, complemented

#7137

C₂

yes

maximal, characteristic, complemented

#3147

C₂²

yes

—

—

C₂²

yes

—

#1848

C₂²

yes

characteristic

#1848

C₂²

yes

—

#1567

—

#1492

—

C₂×#322

C₂²

yes

characteristic

C₂×#135

—

#532

—

C₂²×C₁₆

D₄

yes

—

#495

—

#494

Q₈

yes

characteristic

#493

Q₈

yes

—

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

C₂³

yes

—

—

—

—

—

D₄⋊C₈

—

#322

—

C₂×#41

—

#135

—

#135

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄

yes

—

C₄⋊Q₈

—

#127

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₈

yes

—

#108

—

#108

—

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

D₈

yes

—

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

C₄×C₈

yes

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

#127

yes

characteristic

C₂²×C₄

#137

yes

—

Q₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊Q₈

yes

—

C₂×C₈

#123

yes

—

C₂×C₈

C₂×D₈

yes

—

C₂×C₈

#137

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂²⋊Q₈

yes

characteristic

C₄²

#123

yes

characteristic

C₄²

C₂×D₈

yes

—

C₁₆

—

C₂³

—

C₂³

#497

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

#399

yes

characteristic

105

C₂×C₄

#474

yes

—

106

C₂×C₄

#480

yes

—

107

C₂×C₄

#493

yes

—

108

C₂×C₄

#493

yes

characteristic

109

C₂×C₄

#497

yes

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

#474

yes

—

114

C₈

#480

yes

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

#1722

yes

—

119

C₂²

#2093

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

#1712

yes

characteristic

124

C₄

#1722

yes

characteristic

125

C₄

#2093

yes

—

126

C₂

—

127

C₂

—

128

C₂

#10673

yes

characteristic

129

C₂

#13532

yes

—

	subgroup	G/H
center	C₂×C₄	#493
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#2093
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#493

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂²

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322 ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #12910	C₂	semidirect
4. #8826	C₂	semidirect
5. #7137	C₂	semidirect

H ≅ G/N

type

2. #12910

C₂

semidirect

4. #8826

C₂

semidirect

5. #7137

C₂

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13209	2
#12910	2
#9862	2
#8826	2
#7137	2

subgroup

index

Subgroups — #146239

Conjugacy classes of subgroups

Complements

Maximal subgroups